Accessible fragments of generalized probabilistic theories, cone equivalence, and applications to witnessing nonclassicality

The formalism of generalized probabilistic theories (GPTs) was originally developed as a way to characterize the landscape of conceivable physical theories. Thus, the GPT describing a given physical theory necessarily includes all physically possible processes. We here consider the question of how to provide a GPT-like characterization of a particular experimental setup within a given physical theory. We show that the resulting characterization is not generally a GPT in and of itself, rather, it is described by a more general mathematical object that we introduce and term an accessible GPT fragment. We then introduce an equivalence relation, termed cone equivalence, between accessible GPT fragments (and, as a special case, between standard GPTs). We give a number of examples of experimental scenarios that are best described using accessible GPT fragments, and where moreover, cone-equivalence arises naturally. We then prove that an accessible GPT fragment admits of a classical explanation if and only if every other fragment that is cone equivalent to it also admits of a classical explanation. Finally, we leverage this result to prove several fundamental results regarding the experimental requirements for witnessing the failure of generalized noncontextuality. In particular, we prove that neither incompatibility among measurements nor the assumption of freedom of choice is necessary for witnessing failures of generalized noncontextuality, and moreover, that such failures can be witnessed even when using arbitrarily inefficient detectors.

Figure 1

A series of figures illustrating how the state space of an accessible GPT fragment arises from its associated multisource $P$. In these examples, the gray region demarcates the full set of normalized states in the underlying GPT within which the fragment is embedded. In (a), we depict the atomic preparations, namely, the state vectors describing the output of the multisource for a fixed setting value $x\in X$ and a fixed outcome $a\in A$ of the multisource. In this example, setting $x=0$ leads to two possible outcomes, with the two associated (subnormalized) state vectors depicted in red; similarly for $x=1$ in green. Setting $x=2$ has only a single outcome, so the corresponding state vector (depicted in blue) is normalized and lies in the gray region. (b) We show all of these atomic preparations on a single picture. (c) Even for a fixed setting value, one can access more general preparations by considering outcomes described by a generic effect $e$; that is, by considering outcomes arising from arbitrary postprocessings of the atomic outcomes. In doing so, one can access all the state vectors in the red (for $x=0$), green (for $x=1$), and blue (for $x=2$) regions. Mathematically, these regions are zonotopes. (A zonotope is a convex set constructed by taking some set of vectors $v_i$ and then taking the set of all vectors that can be reached by linear combinations of the form ${\sum }_i{\alpha }_i{v}_i$ such that ${\alpha }_i\in [0,1]$.) Note that every such zonotope has a single vector in the underlying normalized state space (the gray region), and that this vector is generated by the corresponding diagram when $e$ is the unit effect (so that one effectively ignores the outcome of the multisource). (d) The most general states that are achievable for a given multisource can be accessed by considering arbitrary pre and postprocessings, depicted here by the comb $c$ in the diagram. The accessible GPT fragment's state space corresponds to the displayed polytope, which is the convex hull of the zonotopes from (b). This example is illustrative of the generic case. That is, the accessible state space is always given by the convex hull of a set of zonotopes each contained within the full state space, where the maximal element of each zonotope lies in the hyperplane of normalized states. Note, moreover, that any such geometry can be realized by some multisource within the full GPT. Similarly, the accessible effect space is the convex hull of a set of zonotopes each contained within the full effect space where, in this case, the maximal elements are all simply the unit effect. Again we find that any such geometry can be realized by some multimeter within the full GPT. Note that in this example, the subset of normalized states in the fragment forms a strict subset of the normalized states in the fundamental underlying GPT.

Figure 2

Here we depict two distinct three-dimensional accessible GPT fragment state spaces which define identical cones. (a) The first is the inner polytope in dark violet-gray constructed in Fig. 1. (b) The second is the polytope in orange, which happens to contain the dark violet-gray one. (c) The cones defined by these two state spaces coincide. The extremal rays of the cone common to both polytopes are depicted by the protruding red arrows. As in Fig. 1, the gray region demarcates the convex set of normalized states for the full GPT. The black triangle lies within this region, as it corresponds to the set of normalized states in the inner polytope.

Figure 3

(a) The effect space arising if the measurement whose effect space is shown in (b) is implemented inefficiently. (b) The effect space corresponding to a given measurement. (c) The effect space arising if the measurement from (b) is implemented noisily. The cones defined by each effect space are are the regions whose boundaries are denoted by black arrows. Clearly, the effect spaces in (a) and (b) define the same cone, which is distinct from the one defined by the effect space in (c).